The mean value theorem states that if F(x) is a...

The mean value theorem states that if $F(x)$ is a differentiable function on the interval $[a, b],$ then there exists some number $c$ between $a$ and $b$ such that $$F^{\prime}(c)=\frac{F(b)-F(a)}{b-a}$$ Explain why the mean value theorem implies that if a car averages 60 miles per hour in some 10 -minute interval, then the car's instantaneous velocity is 60 miles per hour at least once in that interval.

The mean value theorem states that if $F(x)$ is a differentiable function on the interval $[a, b],$ then there exists some number $c$ between $a$ and $b$ such that
$$F^{\prime}(c)=\frac{F(b)-F(a)}{b-a}$$
Explain why the mean value theorem implies that if a car averages 60 miles per hour in some 10 -minute interval, then the car's instantaneous velocity is 60 miles per hour at least once in that interval.

Key Concepts

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a particular moment in time, represented by the derivative of the position function. The Mean Value Theorem implies that if the average velocity over an interval is a certain value, then at some point during the interval, the instantaneous velocity must equal that average value.

Average Rate of Change

The average rate of change of a function over an interval is calculated as the difference in the function's values at the endpoints, divided by the length of the interval. In the context of motion, it represents the average speed or velocity over that period.

Differentiability

Differentiability is the property of a function implying that its derivative exists at every point in an interval. This is crucial for applying the Mean Value Theorem because it ensures the function behaves smoothly without abrupt changes in slope, allowing for the comparison of average and instantaneous rates.

Mean Value Theorem

The Mean Value Theorem (MVT) states that for any function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the instantaneous rate of change (derivative) equals the average rate of change over the interval. This theorem provides a formal guarantee that average behavior is reflected at some specific moment.

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